One-dimensional random walks with self-blocking immigration

We consider a system of independent one-dimensional random walkers where new particles are added at the origin at fixed rate whenever there is no older particle present at the origin. A Poisson ansatz leads to a semi-linear lattice heat equation and predicts that starting from the empty configuration the total number of particles grows as $c \sqrt{t} \log t$. We confirm this prediction and also describe the asymptotic macroscopic profile of the particle configuration.Comment: Revised version; in particular, details of the proof of the lower bound have been worked out more explicitl

Brownian Web and Oriented Percolation: Density Bounds

In a recent work, we proved that under diffusive scaling, the collection of rightmost infinite open paths in a supercritical oriented percolation configuration on the space-time lattice Z^2 converges in distribution to the Brownian web. In that proof, the FKG inequality played an important role in establishing a density bound, which is a part of the convergence criterion for the Brownian web formulated by Fontes et al (2004). In this note, we illustrate how an alternative convergence criterion formulated by Newman et al (2005) can be verified in this case, which involves a dual density bound that can be established without using the FKG inequality. This alternative approach is in some sense more robust. We will also show that the spatial density of the collection of rightmost infinite open paths starting at time 0 decays asymptotically in time as c/\sqrt{t} for some c>0.Comment: 12 pages. This is a proceeding article for the RIMS workshop "Applications of Renormalization Group Methods in Mathematical Sciences", held at Kyoto University from September 12th to 14th, 2011. Submitted to the RIMS Kokyuroku serie

Disorder relevance for the random walk pinning model in dimension 3

We study the continuous time version of the random walk pinning model, where conditioned on a continuous time random walk Y on Z^d with jump rate \rho>0, which plays the role of disorder, the law up to time t of a second independent random walk X with jump rate 1 is Gibbs transformed with weight e^{\beta L_t(X,Y)}, where L_t(X,Y) is the collision local time between X and Y up to time t. As the inverse temperature \beta varies, the model undergoes a localization-delocalization transition at some critical \beta_c>=0. A natural question is whether or not there is disorder relevance, namely whether or not \beta_c differs from the critical point \beta_c^{ann} for the annealed model. In Birkner and Sun [BS09], it was shown that there is disorder irrelevance in dimensions d=1 and 2, and disorder relevance in d>=4. For d>=5, disorder relevance was first proved by Birkner, Greven and den Hollander [BGdH08]. In this paper, we prove that if X and Y have the same jump probability kernel, which is irreducible and symmetric with finite second moments, then there is also disorder relevance in the critical dimension d=3, and \beta_c-\beta^{ann}_c is at least of the order e^{-C(\zeta)\rho^{-\zeta}}, C(\zeta)>0, for any \zeta>2. Our proof employs coarse graining and fractional moment techniques, which have recently been applied by Lacoin [L09] to the directed polymer model in random environment, and by Giacomin, Lacoin and Toninelli [GLT09] to establish disorder relevance for the random pinning model in the critical dimension. Along the way, we also prove a continuous time version of Doney's local limit theorem [D97] for renewal processes with infinite mean.Comment: 36 pages, revised version following referee's comments. Change of title. Added a monotonicity result (Theorem 1.3) on the critical point shift shown to us by the referee

One-dimensional Voter Model Interface Revisited

We consider the voter model on Z, starting with all 1's to the left of the origin and all 0's to the right of the origin. It is known that if the associated random walk kernel p has zero mean and a finite r-th moment for any r>3, then the evolution of the boundaries of the interface region between 1's and 0's converge in distribution to a standard Brownian motion (B_t)_{t>0} under diffusive scaling of space and time. This convergence fails when p has an infinite r-th moment for any r<3, due to the loss of tightness caused by a few isolated 1's appearing deep within the regions of all 0's (and vice versa) at exceptional times. In this note, we show that as long as p has a finite second moment, the measure-valued process induced by the rescaled voter model configuration is tight, and converges weakly to the measure-valued process 1_{x0.Comment: revised versio

A Monotonicity Result for the Range of a Perturbed Random Walk

We consider a discrete time simple symmetric random walk on Z^d, d>=1, where the path of the walk is perturbed by inserting deterministic jumps. We show that for any time n and any deterministic jumps that we insert, the expected number of sites visited by the perturbed random walk up to time n is always larger than or equal to that for the unperturbed walk. This intriguing problem arises from the study of a particle among a Poisson system of moving traps with sub-diffusive trap motion. In particular, our result implies a variant of the Pascal principle, which asserts that among all deterministic trajectories the particle can follow, the constant trajectory maximizes the particle's survival probability up to any time t>0.Comment: 10 pages, 1 figure. To appear in Journal of Theoretical Probabilit

Universality in marginally relevant disordered systems

We consider disordered systems of directed polymer type, for which disorder is so-called marginally relevant. These include the usual (short-range) directed polymer model in dimension (2+1), the long-range directed polymer model with Cauchy tails in dimension (1+1) and the disordered pinning model with tail exponent 1/2. We show that in a suitable weak disorder and continuum limit, the partition functions of these different models converge to a universal limit: a log-normal random field with a multi-scale correlation structure, which undergoes a phase transition as the disorder strength varies. As a by-product, we show that the solution of the two-dimensional Stochastic Heat Equation, suitably regularized, converges to the same limit. The proof, which uses the celebrated Fourth Moment Theorem, reveals an interesting chaos structure shared by all models in the above class.Comment: 49 pages. Fixed typos, added comments and references. To appear in Ann. Appl. Proba